On Extremal Values of the <i>N<sub>k</sub></i>-Degree Distance Index in Trees

On Extremal Values of the <i>N<sub>k</sub></i>-Degree Distance Index in Trees

The <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>N</mi><mi>k</mi></msub></semantics></math></inline-formula>-index (<i>k</i>-distance degr...

Full description

Saved in:
Bibliographic Details
Main Authors: Zia Ullah Khan, Quaid Iqbal
Format: Article
Language:English
Published: MDPI AG 2025-07-01
Series:Mathematics
Subjects:
distance topological index
<i>N<sub>k</sub></i>-index
trees
broom graph
distances in graphs
Online Access:https://www.mdpi.com/2227-7390/13/14/2284
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:The <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>N</mi><mi>k</mi></msub></semantics></math></inline-formula>-index (<i>k</i>-distance degree index) of a connected graph <i>G</i> was first introduced by Naji and Soner as a generalization of the distance degree concept, as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>N</mi><mi>k</mi></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>d</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></munderover></mstyle><mfenced separators="" open="(" close=")"><mstyle displaystyle="true"><munder><mo>∑</mo><mrow><mi>v</mi><mo>∈</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></munder></mstyle><msub><mi>d</mi><mi>k</mi></msub><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow></mfenced><mi>k</mi></mrow></semantics></math></inline-formula>, where the distance between <i>u</i> and <i>v</i> in <i>G</i> is denoted by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>d</mi><mo>(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo>)</mo></mrow></semantics></math></inline-formula>, the diameter of a graph <i>G</i> is denoted by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>d</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></semantics></math></inline-formula>, and the degree of a vertex <i>v</i> at distance <i>k</i> is denoted by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><msub><mi>d</mi><mi>k</mi></msub><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mo>{</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo>∈</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mspace width="0.166667em"></mspace><mspace width="4pt"></mspace><mi>d</mi><mrow><mo>(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo>)</mo></mrow><mo>=</mo><mi>k</mi><mo>}</mo></mrow></mrow></semantics></math></inline-formula>. In this paper, we extend the study of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>N</mi><mi>k</mi></msub></semantics></math></inline-formula>-index of graphs. We introduced some graph transformations and their impact on the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>N</mi><mi>k</mi></msub></semantics></math></inline-formula>-index of graph and proved that the star graph has the minimum, and the path graph has the maximum <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>N</mi><mi>k</mi></msub></semantics></math></inline-formula>-index among the set of all trees on <i>n</i> vertices. We also show that among all trees with fixed maximum-degree <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Δ</mo></semantics></math></inline-formula>, the broom graph <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>B</mi><mrow><mi>n</mi><mo>,</mo><mo>Δ</mo></mrow></msub></semantics></math></inline-formula> (consisting of a star <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>S</mi><mrow><mo>Δ</mo><mo>+</mo><mn>1</mn></mrow></msub></semantics></math></inline-formula> and a pendant path of length <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>−</mo><mo>Δ</mo><mo>−</mo><mn>1</mn></mrow></semantics></math></inline-formula> attached to any arbitrary pendant path of star) is a unique tree which maximizes the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>N</mi><mi>k</mi></msub></semantics></math></inline-formula>-index. Further, we also defined and proved a graph with maximum <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>N</mi><mi>k</mi></msub></semantics></math></inline-formula>-index for a given number of <i>n</i> vertices, maximum-degree <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Δ</mo></semantics></math></inline-formula>, and perfect matching among trees. We characterize the starlike trees which minimize the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>N</mi><mi>k</mi></msub></semantics></math></inline-formula>-index and propose a unique tree which minimizes the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>N</mi><mi>k</mi></msub></semantics></math></inline-formula>-index with diameter <i>d</i> and <i>n</i> vertices among trees.
ISSN:2227-7390

Similar Items